From the side of response, the impact of non-Herminity on observable has been studied using field-theoretical techniques within the linear response theory(e.g. the Kubo formula) [1]}, [2]}, [3]}, [4]}, [5]}. They found that there is no link between non-Hermitian topological invariants and the quantization of observables [1]} and the observables are no longer quantized in general otherwise requiring more strict conditions than a nonzero non-Hermitian Chern number [2]}. In these studies, the non-Hermitian term was introduced via self-energy in the low-energy limit [3]}, or phenomenologically introduced to add into the system, so a non-Hermitian version of the TKNN can be derived to show the topological contribution [2]}. This gives rise a question that how the decay rate depends on the momentum of the electron, and how the decay depends on the couplings between the system and the environment, and how a non-Hermitian system with \(k-\) dependent decay rate response to an external stimulus.
